AcademyFluids
Academy
Density in Gases and Liquids
Level 1 - Physics topic page in Fluids.
Principle
Density measures how much mass occupies a given volume.
Notation
\(\rho\)
mass density
\(\mathrm{kg\,m^{-3}}\)
\(m\)
mass
\(\mathrm{kg}\)
\(V\)
volume
\(\mathrm{m^{3}}\)
\(dm\)
small mass element
\(\mathrm{kg}\)
\(dV\)
small volume element
\(\mathrm{m^{3}}\)
Method
For a uniform sample, density is one ratio; for a nonuniform fluid, it must be defined locally.
Average density
\[\rho=\frac{m}{V}\]
Rearrange sample
\[m=\rho V\]
Use this when the density is effectively uniform across the sample.
Local density
\[\rho=\frac{dm}{dV}\]
This is the definition when density changes with position.
Recover total mass
\[m=\int_V \rho\,dV\]
Liquids are often modeled with nearly constant density, while gases change density more readily when pressure or temperature changes.
Rules
These are the compact density relations.
Average density
\[\rho=\frac{m}{V}\]
Mass from volume
\[m=\rho V\]
Volume from mass
\[V=\frac{m}{\rho}\]
Local density
\[\rho=\frac{dm}{dV}\]
Examples
Question
A liquid has density
\[840\,\mathrm{kg\,m^{-3}}\]
Find the mass in \[2.5\times10^{-3}\,\mathrm{m^3}\]
Answer
\[m=\rho V=840(2.5\times10^{-3})=2.10\,\mathrm{kg}\]
Checks
- Density units are mass per volume, not mass per area.
- A larger density means more mass in the same volume.
- Use the local definition when density varies through the fluid.
- Liquids are usually modeled as less compressible than gases.